DoH only really poses an issue if you've got split-horizon DNS (i.e. there's also a public facing record).
Firefox (for example) will attempt to use DoH, but if resolution fails that way it'll fall back to local DNS (FF also has a canary domain you can use to disable DoH).
Chrome's even simpler, it just tries DoH to your system configured resolver (and falls back if DoH can't be done).
Basically, you can just DHCP your DNS as normal and it all sort of works